[hamdani yusuf (OP)]

This is what I find when googling "kinetic energy in temperature"

  Temperature is a measure of the average kinetic energy of all the molecules in a gas. As the temperature and, therefore, kinetic energy, of a gas changes, the RMS speed of the gas molecules also changes. The RMS speed of the molecules is the square root of the average of each individual velocity squared.

In wikipedia,
https://en.m.wikipedia.org/wiki/Kinetic_theory_of_gases

The kinetic theory of gases is a historically significant, but simple, model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles (atoms or molecules), all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles and, thus, the nature of kinetic energy transfers during collisions is strictly thermal.   

average kinetic energy per molecule of ideal gas.

For polyatomic gases, more energy is required to increase the temperature by the same amount. It means some amount of kinetic energy don't contribute to the increase of temperature.
The measurement of temperature should exclude contribution of the system's total translational and angular momentum. It should also exclude rotational and vibrational movement of individual particle.
The question thus becomes, what kind of movements are left to constitute the measure of temperature?

[Bored chemist]

The equipartition principle says that , once the system is at equilibrium, the energy is shared equally among all the possible types- radiation, vibration, translation, electronic- etc.
So you can choose any of them as a measure of temperature.

https://en.wikipedia.org/wiki/Equipartition_theorem

For some systems- far from equilibrium- the temperatures can be different- or even negative.

[hamdani yusuf (OP)]

For some systems- far from equilibrium- the temperatures can be different- or even negative.

How is the temperature measured before reaching equilibrium?
Imagine ten giant molecules each forming 1 kg of metal balls contained in 1 cubic meter box floating in the space. They don't move relatively to each other nor to the box but rotate at 1 rps. Do their angular speeds affect the measure of the system's temperature?

[Bored chemist]

How is the temperature measured before reaching equilibrium?

Sometimes you can do this spectroscopically.
Sometimes it's impossible.

Do their angular speeds affect the masure of the system's temperature?

Hypothetically, yes.
There's a school experiment "measuring the temperature of a flame"- it's not very accurate but...
You take a lump of metal (If i remember rightly we used a large hexagonal nut) and heat it in boiling water so it reaches 100C
And then you put it in a beaker containing 100 ml of water at zero degrees  and you measure the temperature rise of the water.
Then you heat the same lump of metal in the  flame until it gets as hot as the flame (that's an approximation but...).
Then you put it into a beaker with 100 ml of ice cold  of water and measure the temperature rise.
The temperature of the flame can be calculated from the ratio of the temperature changes etc.

It's not a very accurate measurement but it illustrates the point.

You can measure the temperature of your  system of a box of  metal balls floating in space by filling the box with water.

Viscous drag will slow the balls down and warm the water up.

And that lets you calculate the initial temperature of the box + contents.

[hamdani yusuf (OP)]

You can measure the temperature of your  system of a box of  metal balls floating in space by filling the box with water.

Viscous drag will slow the balls down and warm the water up.

That would convert mechanical kinetic energy into thermal energy. In Joule's experiment, it was accompanied by increase of temperature. It means that the end temperature is different than initial temperature,  which is the one we want to determine.

[Bored chemist]

That would convert mechanical kinetic energy into thermal energy.

Is there a difference?

It means that the end temperature is different than initial temperature,  which is the one we want to determine.

So, it is exactly the same  as the experiment to measure the temperature of a flame,

[hamdani yusuf (OP)]

That would convert mechanical kinetic energy into thermal energy.

Is there a difference?

It means that the end temperature is different than initial temperature,  which is the one we want to determine.

So, it is exactly the same  as the experiment to measure the temperature of a flame,

an object with high mechanical energy but low thermal energy is cool.
an object with low mechanical energy but high thermal energy is hot.

[Bored chemist]

That would convert mechanical kinetic energy into thermal energy.

Is there a difference?

It means that the end temperature is different than initial temperature,  which is the one we want to determine.

So, it is exactly the same  as the experiment to measure the temperature of a flame,

an object with high mechanical energy but low thermal energy is cool.
an object with low mechanical energy but high thermal energy is hot.

But, as we have seen, if the "molecules" are a Kg, the distinction isn't clear.
In a more realistic scenario, the spheres wouldn't be  stationary in their box- the uncertainty principle forbids it- so, from time to time they would bump into each other and they would convert rotational energy into translational and vibrational energy.
After a while, they would reach equilibrium where, on average, their energy is divided equally among all the available degrees of freedom.
And that's a temperature.

[hamdani yusuf (OP)]

In a more realistic scenario, the spheres wouldn't be  stationary in their box- the uncertainty principle forbids it- so, from time to time they would bump into each other and they would convert rotational energy into translational and vibrational energy.
After a while, they would reach equilibrium where, on average, their energy is divided equally among all the available degrees of freedom.
And that's a temperature.

I think it's possible to stabilize the system for a long time before the balls bump into each other, which makes the equilibrium has to wait even longer. Does it mean that the system's temperature can't be defined yet?

[Bored chemist]

You can have a system with more than one temperature.

[hamdani yusuf (OP)]

You can have a system with more than one temperature.

What if all of those balls rotate at the same angular speed? How many temperatures are there?

[Bored chemist]

Some distributions do not correspond to a temperature.
The best you could do would be , as I said, to average the energy somehow, for example by adding water.
Even a little gas would do, if you waited long enough.

[hamdani yusuf (OP)]

Let's have another case to contrast thermal with mechanical energy.
Box A contains 1 mole of Hydrogen gas (H2) at STP.
Box A contains 1 mole of Helium gas (He) at STP.
Both of them are sealed from environment.
A rotating ball with 1kJ of rotational energy is introduced into each of those containers.
When the balls stop spinning, their kinetic energies have been transfered to those gases. But since their Molar heat capacities are different, their final temperature would be different.   (H2)=28.836 J/(mol·K)   (He)=20.78 J/(mol·K)
Helium will have higher temperature than Hydrogen, despite having higher mass.
It means that Helium will have higher thermal energy than Hydrogen. I conclude that in Hydrogen, there are some kinetic energy which are not converted into thermal energy.

[Bored chemist]

That's correct.
Some of the heat in the hydrogen is in the form of vibrational or rotational energy of the molecules.

[hamdani yusuf (OP)]

It doesn't necessarily imply that vibration and rotation don't contribute to the reading of temperature. It only means that in case of gases,  they contribute less than translation.
In case of solids, the temperature is almost exclusively determined by vibrational motion.
This brings us back to the original question, what is temperature? Kinetic gas theory suggests that it's a form of kinetic energy. But heat transfer shows that it behaves like potential energy. The energy flow is similar to how electric batteries behave when connected to other batteries in parallel. Or how the liquid flow in connected vessels.

[Bored chemist]

It only means that in case of gases,  they contribute less than translation.

They all contribute the same  energy per degree of freedom.

[hamdani yusuf (OP)]

Is it possible for a system to have a non-integer degree of freedom? If it is, what does it mean?

[Bored chemist]

Sort of, because the energy levels are quantised.
For a cold diatomic gas like nitrogen there's not enough to excite vibrations and so the only contributions to the heat capacity are translation and rotation.
That's 3 translational degrees and 2 rotational ones (rotation about the axis between the centres of the two atoms doesn't count) making 5 in total
When the gas is hot there is enough energy to get the molecules vibrating and that adds some more degrees of freedom into which energy can be placed.

That adds another degree of freedom , making 6 in total.

So the calculated heat capacities under those conditions are 2.5kT and 3 kT

But, at an intermediate temperature  some, but not all, of the molecules will have enough energy to induce vibrations.
And under those conditions the average number of degrees of freedom will be somewhere between 5 and 6.

[hamdani yusuf (OP)]

Let's consider the molar heat capacities of noble gases which are almost uniform and very close to 21 J/(mol.K).
But their atomic mass are very different, (4 for Helium and 222 for Radon)
Their densities also vary significantly (0.1786 g/L for Helium and 9.73 g/L for Radon).

Their temperatures are almost entirely determined by their average translational motion.
Increasing one mole of Helium by 1 Kelvin increases its internal energy by 20.78 Joule. Since the kinetic energy is ½mv², the average velocity of Helium atoms would increase by √(2 * 20.78 / 4 * 1000) = 101.93 m/s
For Radon, the internal energy will increase by 20.786 Joule.
The average atom velocity would increase by √(2 * 20.786 / 222 * 1000) = 13.68 m/s

[Bored chemist]

That's consistent with the speeds of sound in gases; the heavy gases have lower speeds of sound.

But their atomic mass are very different, (4 for Helium and 222 for Radon)
Their densities also vary significantly (0.1786 g/L for Helium and 9.73 g/L for Radon).

And the ratios are nearly constant
4/ 0.1786=22.40 litres per mole
222/ 9.73=22.82 litres per mole